Ncert Exercises & Solutions

A mass \(M\) is attached to a spring system as shown in the figure. If the mass is displaced from its equilibrium position and then released, what is the time period of its oscillation?

1. \(2\pi \sqrt{\dfrac{M}{k}} \)
2. \(2\pi \sqrt{\dfrac{M}{2k}} \)
3. \(2\pi \sqrt{\dfrac{M}{4k}} \)
4. \(2\pi \sqrt{\dfrac{2M}{3k}} \)

Hint: \(F=-kx\)

Step 1: Find the initial deformation \((2x_0)\) of the spring.
\(Mg=k(2x_0)\)
\(\Rightarrow x_0=\dfrac{Mg}{2k}\)
Step 2: Find the angular frequency of spring-mass system.
\(\dfrac{1}{2}k(x_0+2A)^2=\dfrac{1}{2}Mv^2+\dfrac{1}{2}kx_0^2+MgA\)
\(\Rightarrow k(x_0+2A)^2=M(A\omega)^2+kx_0^2+2MgA\)
\(\Rightarrow kx_0^2+4kA^2+4kAx_0=MA^2\omega^2+kx_0^2+2MgA\)
\(\Rightarrow 4kA^2+4kAx_0=MA^2\omega^2+2MgA\)
\(\Rightarrow 4kA^2+4kAx_0=MA^2\omega^2+4kx_0A~~~[\because Mg=2kx_0]\)
\(\Rightarrow 4kA^2=MA^2\omega^2\)
\(\Rightarrow \omega=\sqrt{\dfrac{4k}{M}}\)
Step 3: Find the time period of spring-mass system.
\(\omega=\dfrac{2\pi }{T}\)
\(\Rightarrow \sqrt{\dfrac{4k}{M}}=\dfrac{2\pi}{T}\)
\(\Rightarrow T=2\pi \sqrt{\dfrac{M}{4k}} \)
Hence, option (3) is the correct answer.

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